Fourier series criteria for operator decomposability

Let U be an invertible operator on a Banach space Y. U is said to betrigonometricallywell-bounded provided the sequence {Uⁿ}_n=– is the Fourier-Stieltjes transform of a suitable projection-valued function E(·): [0, 2](Y). This class of operators is known to apply naturally to a variety of classical phenomena which exclude the presence of spectral measures. In the case Y reflexive we use the Cesáro means _n(U, t) of the trigonometric series _k0 k^–e^iktU^k, whichformally transfers the discrete Hilbert transform to Y, in order to give three separate necessary and sufficient conditions for U to be trigonometrically well-bounded. One of these conditions is sup {_n(U,t): n 1, t [0,2]} <